Page:Great Neapolitan Earthquake of 1857.djvu/191

3rd. In the case of a right circular solid cylinder, fractured from its horizontal base.

Here $\beta =D$, the diameter of the cylinder,

$k^{2}={\frac {5}{16}}\beta ^{2}$;

therefore

${\begin{cases}V={\frac {5}{8}}g\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\\or\\V=20.12{\frac {\mathrm {LD} }{\alpha ^{2}}}\end{cases}}$

(XXV.)

4th. In the case of a hollow parallelopiped, fractured from its horizontal base.

Assuming the thickness of the walls small, as before, and $\gamma$ being that side of the parallelopiped, which is the axis of inceptive rotation,

$k^{2}={\frac {\beta ^{2}(\beta +2\gamma )}{4(\beta +\gamma )}}$;

therefore

${\begin{cases}V={\frac {1}{2}}g\times {\frac {\mathrm {L} \beta (\beta +2\gamma )}{\alpha ^{2}(\beta +\gamma )}}\\or\\V=16.10\times {\frac {\mathrm {L} \beta (\beta +2\gamma )}{\alpha ^{2}(\beta +\gamma )}}\end{cases}}$

(XXVI.)

5th. In the case of a hollow square prismatic tower, fractured from its base.

Then, $\beta =\gamma$, and (Eq. XXVI.) becomes

${\begin{cases}V={\frac {3}{4}}g\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\\or\\V=24.15\times {\frac {\mathrm {L} \beta }{\alpha ^{2}}}\end{cases}}$

(XXVII.)